The peer-review process is old. I wonder when anonymous peer review appeared and more importantly what the motivations behind it were (vs. non-anonymous peer review).
I believe that in the 1930s, peer-review was mostly done directly by editors-as-peers. This belief is based on this letter by Albert Einstein to the Physical Review, 1936:
Dear Sir, We (Mr. Rosen and I) had sent you our manuscript for publication and had not authorized you to show it to specialists before it is printed. I see no reason to address the in any case erroneous comments of your anonymous expert. On the basis of this incident I prefer to publish the paper elsewhere.
P.S. Mr. Rosen, who has left for the Soviet Union, has authorized me to represent him in this matter.
This seems to indicate that Einstein, who publied about 300 papers in the first half of the 20th century, was unfamiliar with anonymus peer-reviewing (for the sake of the story, it turned out the reviewer's opinion was right on this matter).
Nature documents the introduction of a formal, anonymous, systematic peer-review system in 1967, mainly to deal with the accumulation of unpublished-yet-not-rejected papers. Before this, reviewing was performed by editors, eventually seeking for advice outside.
From what I have read on the subject, it seems to me that non-anonymous peer review never really existed: today's peer review is derived from reviewing comities in publications, that were allowed to evolve into more flexible, distributed reviewing groups in the 60s/70s because both of the possibility of doing it (Xerox copier comes to mind) and the necessity of doing it (to face accumulation of paper, as stated in Nature example above). Nobody ever thought of giving the name of the reviewers, are they were not considered important. There are trends today to make the reviewing process more anonymous (double-blind, the reviewer doesn't know who is the author) and less anonymous (disclosing the name of the reviewers), but I don't believe the system will change in a close future, as there is no consensus in the scientific community about this.